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Physical Review A

Quantum-state tomography for optical polarization with arbitrary photon numbers

A. B. Klimov, José Luis Romero, Sascha Wallentowitz

Published Feb 28, 2014• 8 citations
Publisher

Abstract

A scheme for quantum-state tomography is presented that can be performed for polarized light with an arbitrary photon-number distribution. The proposed method fills the gap between existing polarization-tomography schemes for single-photon states and for optical fields with very large photon numbers. It consists of an optical homodyne setup triggered by the outcome of a single-photon counting module. The configuration space of polarization is parametrized by Euler angles and by an additional interference parameter, whose assessment requires conditional homodyne measurements.

DOI
10.1103/physreva.89.020101

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A scheme for quantum-state tomography is presented that can be performed for polarized light with an arbitrary photon-number distribution. The proposed method fills the gap between existing polarization-tomography schemes for single-photon states and for optical fields with very large photon numbers. It consists of an optical homodyne setup triggered by the outcome of a single-photon counting module. The configuration space of polarization is parametrized by Euler angles and by an additional interference parameter, whose assessment requires conditional homodyne measurements.
Quantum state tomography is an essential tool for characterizing quantum systems and underpins nearly every experimental realization of quantum technologies. Conventional tomography relies on performing a sequence of projective measurements on many identical copies of a quantum state, requiring the measurement apparatus to be reconfigured between successive acquisitions. As the Hilbert-space dimension increases, the number of required measurements grows rapidly; in practice, additional overcomplete measurements are often performed to improve robustness to experimental imperfections. Here, we introduce a tomography platform based on structured anisotropic media that performs informationally complete measurements of photonic polarization qubits within a single acquisition. The approach employs three liquid-crystal metasurfaces with spatially varying optic-axis orientations that transform the input polarization into a far-field distribution of discrete transverse-momentum modes. Each diffraction pattern uniquely determines the polarization state, enabling its reconstruction without sequential changes to the measurement apparatus. Unlike previous implementations, our scheme is intrinsically photon-number independent: the same optical device operates identically for arbitrary photon numbers, while the desired photon-number sector can be selected afterwards through post-selection of the corresponding nnn-fold coincidence events. We experimentally demonstrate single-frame quantum state tomography of both single- and two-photon polarization states, providing a simple and scalable route toward efficient quantum-state characterization.
Quantum state tomography (QST), the process through which the density matrix of a quantum system is characterized from measurements of specific observables, is a fundamental pillar in the fields of quantum information and computation. In this work, we propose a simple QST method to reconstruct the density matrix of two qubits encoded in the polarization and path degrees of freedom of a single photon, which can be realized with a single linear-optical setup. We demonstrate that the density matrix can be fully described in terms of the Stokes parameters related to the two possible paths of the photon, together with a quantum version of the two-point Stokes parameters introduced here. Our findings put forward photonic circuits for the investigation of the dynamics of open quantum systems.
We have reconstructed the quantum state of optical pulses containing single photons using the method of phase-randomized pulsed optical homodyne tomography. The single-photon Fock state 1> was prepared using conditional measurements on photon pairs born in the process of parametric down-conversion. A probability distribution of the phase-averaged electric field amplitudes with a strongly non-Gaussian shape is obtained with the total detection efficiency of (55+/-1)%. The angle-averaged Wigner function reconstructed from this distribution shows a strong dip reaching classically impossible negative values around the origin of the phase space.
Quantum state tomography is used in quantum communication, computing, and cryptography to characterize quantum states enabling assessment and control of quantum systems. Rapidly scaling quantum technologies require fast, accurate, and simple techniques, but current tomography methods are bulky, hard to scale, and slow. To overcome these limitations, we experimentally demonstrate Fourier tomography [1], of single-photon and entangled photon pairs generated by a quantum dot. As shown in Fig. 1(a) for two-photon tomography, the setup consists of a continuously rotating quarter waveplate, a fixed linear polarizer and a single photon detector per qubit. As the waveplates rotates with different speed, we record the coincidences rate between the two detectors. This signal can be expressed as a Fourier series whose coefficients can be related to the polarization state of light. Fig. 1(b) shows the simulated coincidence rate as the waveplate rotates for input Bell states∣Φ+⟩\vert \Phi^{+}\rangle∣Φ+⟩and∣Φ−⟩\left\vert\Phi^{-}\right\rangle∣Φ−⟩. By capturing the coincidences variations as a Fourier series through a single rotating element and fixed polarizer, we simplify the tomography process, providing an accurate, and simple approach suitable for scalable quantum systems.
We propose and demonstrate a method for quantum-state tomography of qudits encoded in the quantum polarization of NNN-photon states. This is achieved by distributing NNN photons nondeterministically into three paths and their subsequent projection, which for N=1N=1N=1 is equivalent to measuring the Stokes (or Pauli) operators. The statistics of the recorded NNN-fold coincidences determines the unknown NNN-photon polarization state uniquely. The proposed, fixed setup manifestly rules out any systematic measurement errors due to moving components and allows for simple switching between tomography of different states, which makes it ideal for adaptive tomography schemes.
Summary form only given. The single photon Fock state is one of the most fundamental states of the light field. It is highly non-classical and reveals the wave-particle duality of light most strikingly. Its marginal distributions are of non-Gaussian shape and its Wigner function exhibits a strong negativity around the origin of phase space. Although the reconstruction of the Fock state has already been performed for the motional state of trapped beryllium ions, in the optical domain this task has not been resolved so far, the main difficulty being the lack of coherent single-photon sources. We present results of applying the technique of optical homodyne tomography to the single photon Fock state prepared by conditional measurements on a photon pair produced in parametric down conversion.